Before you start
Find a percentage of an amount and convert fractions such as 1/4 into percentages. All money in an individual question uses one currency. These are simplified school models; include extra costs only when the question states them.
Need a starting point? Percentage word problems with step-by-step answers builds a related prerequisite or visual idea.
Cost price, selling price, profit and loss
Cost price, written CP, is the amount paid to buy the item in the question. Selling price, written SP, is the amount received when it is sold. If SP exceeds CP, profit = SP − CP. If SP is below CP, loss = CP − SP. If they are equal, there is neither profit nor loss.
Keep the amount and percentage separate. “₹100 profit” names a money difference. “25% profit” compares that difference with the original cost. Two items can earn the same amount of profit but have different profit percentages because their costs differ.
Why the denominator is cost price
In this school convention, cost price is the starting 100%. Profit percentage = profit ÷ CP × 100. Loss percentage = loss ÷ CP × 100. Circle the cost in the question before substituting. This prevents accidentally dividing by the selling price.
A business may also report a margin, which uses selling price as its denominator. That is a different measure. Do not replace a school profit-percentage answer with a margin calculation unless the question explicitly asks for margin. The word “percentage” needs a base.
Find selling price from a stated percentage
A 15% profit means the selling price is 115% of cost: SP = CP × 1.15. A 15% loss means SP is 85% of cost: SP = CP × 0.85. You can instead calculate the gain or loss first and then add or subtract it. Both methods express the same comparison.
Use a quick check before detailed calculation. Profit should make SP larger than CP; loss should make it smaller. A negative selling price cannot arise from the ordinary loss examples here. The loss-rate multiplier method assumes the stated rate lies between 0% and 100%.
Work backwards to find cost price
If an item sells for ₹690 at a 15% profit, ₹690 represents 115% of its cost, not 100%. Divide by 1.15 to get CP = ₹600. Subtracting 15% of ₹690 uses the wrong base and does not undo the original increase.
For a 20% loss, SP is 80% of CP. Hence CP = SP ÷ 0.80. Check a reverse calculation by applying the stated rate to the recovered cost and confirming that it gives the supplied selling price.
Extra expenses and marked price
When a question explicitly includes transport or repair as part of the purchase cost, add those expenses before finding the gain. For example, ₹800 purchase plus ₹200 stated repairs gives ₹1,000 total cost for that problem. Do not invent costs that were not supplied.
Marked price is a label or advertised price; discount is calculated from that marked price. Selling price is what the buyer actually pays after the stated discount. A discount does not automatically mean a loss: you must still compare the final selling price with cost.
See the relationships
Read the labels and explanation together. These diagrams illustrate the examples stated in their captions; the practice questions give their own values.
Worked examples, step by step
Example 1: Find the profit percentage when CP = ₹400 and SP = ₹500.
- Profit = 500 − 400 = ₹100.
- Use CP as the base: 100 ÷ 400 × 100 = 25%.
- Check: 25% of ₹400 is ₹100, so SP = 400 + 100 = ₹500.
Example 2: An item costs ₹750 and sells for ₹660. Find loss and loss percentage.
- Loss = 750 − 660 = ₹90.
- Loss percentage = 90 ÷ 750 × 100 = 12%.
- Check: 88% of ₹750 is ₹660.
Example 3: Find SP for a ₹600 item sold at 15% profit.
- 15% of ₹600 is ₹90.
- Add the gain: 600 + 90 = ₹690.
- Equivalently, 600 × 1.15 = ₹690.
Example 4: An item sells for ₹720 at a 20% loss. Find CP.
- A 20% loss leaves 80% of cost.
- CP = 720 ÷ 0.80 = ₹900.
- Check: loss = ₹180; 180 ÷ 900 × 100 = 20%.
Example 5: A bicycle costs ₹800, needs ₹200 of repairs, then sells for ₹1,250. Find profit percentage including the stated repairs.
- Total cost = 800 + 200 = ₹1,000.
- Profit = 1,250 − 1,000 = ₹250.
- Profit percentage = 250 ÷ 1,000 × 100 = 25%.
Example 6: An item marked ₹1,000 is discounted 10%. Its cost was ₹750. Is the sale a loss?
- Discount = ₹100, so SP = ₹900.
- Compare with CP: 900 − 750 = ₹150 profit.
- Profit percentage = 150 ÷ 750 × 100 = 20%. A discount and a loss are different comparisons.
Common mistakes and how to repair them
Dividing by SP
For the school profit/loss percentage, use CP in the denominator. State the base in words if unsure.
Undoing a profit by subtracting the same percentage of SP
A 20% rise and a 20% fall use different starting amounts. Recover the original by dividing by the multiplier.
Treating all discounts as losses
Discount compares marked price with selling price. Profit compares selling price with cost price.
Practise before opening the answers
Write your working on paper. Open each answer after making an attempt, then compare the explanation. For construction questions, use the named physical tools.
CP ₹200; SP ₹250. Find profit percentage.
Show answer and reasoning
25%. Profit is ₹50; 50 ÷ 200 × 100 = 25%.
CP ₹500; SP ₹450. Find loss percentage.
Show answer and reasoning
10%. Loss is ₹50; divide by the ₹500 cost.
Find SP for CP ₹800 and 15% profit.
Show answer and reasoning
₹920. 15% of ₹800 is ₹120; add it to cost.
Find SP for CP ₹1,200 and 25% loss.
Show answer and reasoning
₹900. 75% remains, so 1,200 × 0.75 = 900.
SP is ₹960 at 20% profit. Find CP.
Show answer and reasoning
₹800. 960 represents 120% of cost; 960 ÷ 1.20 = 800.
SP is ₹680 at 15% loss. Find CP.
Show answer and reasoning
₹800. 85% remains; 680 ÷ 0.85 = 800.
Two items each earn ₹50 profit. Their costs are ₹100 and ₹250. Compare their profit percentages.
Show answer and reasoning
50% and 20%, respectively. The same money gain is divided by different costs.
A ₹600 item is marked ₹800 and sold with a 10% discount. Find profit percentage.
Show answer and reasoning
20%. SP = 800 × 0.90 = ₹720. Profit ₹120 divided by cost ₹600 gives 20%.
Questions students and parents ask
Can profit percentage exceed 100%?
Yes. If a ₹100 item sells for ₹250, its ₹150 profit is 150% of cost. This is different from saying that more than 100% of the selling price is profit.
What if CP and SP are equal?
Profit and loss are both zero. With a positive cost, the percentage is 0%.
What if the cost price is zero?
The usual percentage formula divides by cost, so it is undefined when CP = 0. A money gain may still be described, but these school percentage formulas require positive cost.
Is this an official NCERT exercise solution page?
No. These are original explanations and practice based on BananaLearn’s course. The linked NCERT Exemplar is a topic reference; textbook editions and chapter numbering can differ.
A simple way to practise together
Use an imaginary shop with two price cards: what we paid and what we received. Ask the learner to explain the money difference first, then what amount counts as 100%. Keep real purchasing decisions separate from this simplified maths model.
Keep learning and practising
For a change of pace, explore our learning games. The games list explains the skills each one practises; not every guide has a matching game level. Return to a fresh question here to check this specific idea.
About this guide
Adapted for independent reading from BananaLearn’s Grade 7 teaching material. We reused our course visual library where it fits and created supporting diagrams for the written explanations. The examples and practice are original learning resources, not official NCERT solutions or reproduced exam questions. School sequences and textbook editions differ.
Topic reference: NCERT Exemplar: Comparing Quantities.
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